SINES. 



and, by fubtraftion,'we have 



,■ _ , — A (» — u) + B (y' - n') + C (v' — u") + &c. '^"^^'^ '"'•° "^''^ expreflions ; ailuming, therefore, x and a aa 

 , , \. ,,..,,' two arcs, whofe tangents are / and /!, we may allume 



Dividing liere both fides by ^ — u, and after the divifion 

 making « = _y, it will become 



for the fines, that only the odd powiru of tiie tangent can 



and : 



(0 



X — z 



tange 



X = At + Bl^ + Cl' + Sec. 



z = A/' + Bt" + C/" + Sec. 

 \ + SBy' + sCy* + yD y'' + Sec. Q) and, by fubtraftion, we ftiaU have 



In order to ellablifli an identity, we muft afcertain what •'^ - * = A {<-<') + B (<'-/<:) + C (/» - t'') + &c. 

 tlic firll member becomes, on the fuppofition of a = _y, after Dividing both members by / — /', and afterwards makint' 

 dividing by j* — u. Now / =: /', we have 



'j^u = fm.T^^. = 2 fin. i (X -\~coi. 4 (* + .)• }~! = ^+ i^^' + SCt^+^Dr+Scc. (0) 



Subftituting, in the latter denominator, for 2 fin. 5 ( .v — z), Alfo, to eftabliih an identity, we have 

 Its value, as exhibited in No. (XVII.), and dividing both 

 its members by .r — z ; that frattion becomes 



z)' + Sec. [ cof. ^{x + z); 

 3-4 J 



and if now we make .v = z, we (hall find the refult will be 

 I 



L 2. •? .4 



cof. X 

 I 



Now 



cof. 



,/(i-fin.^x) v{^-y) 



= (•-/)- 



But fin. Ix — z) = (x — z) (x 



2 . ^ 



+ —- (x - z)' - Sec. 



2 ■ 3 •4-5 



0' 



If, therefore, we fiibftitute this value for fin. (.v — z), 

 y' 'I ■} I'?.? i^.?7 in the denominator of the latter fradtion, it becomes, after 



i + - + — y' + T^f'y'+ T^^iy' + ^'^^ dividing both members by (.V - z), 



,4' 2 . 4 . " 2.4 



and confequently this feries is identical with equation (S), 

 viz. with 



A + i'E-y' + sCy^ 4 i^y' + 9'E-y' + &c- 

 Comparing the homologous terms, we have 



A = I ; B 



X — z 



cof. X . cof. 



+ 



Z)3 



2.3.4.5 



— &C. 



2.3 2. 4. J 2.4.6.7 



■7- ; &c. 



which, on the fuppofition we have made of / ^ /' ; and, 

 confequently, x = z, reduces to 



3f.^ 



carrying, therefore, thefe values into the equation {■>), and 

 fubftituting fin. x for j, we have 



fin.'.v i.3fin.''j.- i.3.5fin.'j' , ,,,,,, 



.■« = fin.,v+ +—^ +—S-^ +&C.XIX.] 



2.3 2.4.5 2.4.6.7 



I I 



1 + tan.-.v ~ 1 -I- /* 

 ■ t" + t' - Sec. 



Icc.'x 



I _ /^ -f. /•> 



which is therefore identical with our equation (5), t'l'z. 



A + 3Br + 5C/' + 7n'' + &c. 



whence, by comparing the coefficients of the like powers 

 and fince the fin. x= cof. (-'- — .vj, we have, by fubfti- of'' we have 



^ . . . A = I, B = - „ C = ,', D = -- J, &c. 



luting this expreflion for fin. x, in the above feries, , . , , r 1 ,i- i , , ■ 



winch values tubititutcu ui equation (;), gives 



iT ^ cof.'x l.3Cof.^x i.3.ccof.'« . ,.-,,. 



•c — _ _ cof. X — ' ■ =^ A' = t~n ■*' — i tan.' x- -|- i tan. ' .v — .' 



2 ■ 2.3 2.4.5 2.4.6.7 tan.'x + &c. (XXI.) 



-Sec. (XX.) in winch the arc is cxprefl'ed in terms of its We fliall „ow conclude this article by exhibiting, in a 



coline. connected order, a few of tlie moll im];ortai;t developemcnts 



In order to invelligate a formula, which fliall exprefs of angular funiliions ; obferviiig that A, B, C, &c. are tU'i 



the arc in terms ol its tangent, it may be fliewn, as before, preceding terms. 



{in.- 11 . 3- fin. -a _ c'fin.'a _, 7' fin. a 



2-3 4-S 6-7 8.9 



D h &c. 



where A, B, C, Sec. are tlie preceding terms. 



2. fin. a z= ,1 — A + 



2-3 4 



0^, - - »•;-, " 



3. .-J = 90° — 



4. cof. a = I 



.r. rt 



cof.' a 3 cof.' a 3-5 e"''.' n 



2.3 2.4.5" 2^4 •6-7 



A - -^^ B - -- C - -' 



3-4 5-6 7 



- &c. 



D 



&c. 

 C 2 



J. a 



